We show in a first step that X0
X−1 =
X2
X1 holds only in the special case ofµD= 1.
This implies that in all other cases the economy is not in the post-disinflation state characterized by perpetual oscillations immediately after the policy is ap- plied. The second step is to show that there is no convergence process of the ratio Xt
Xt−1
= Xt+2
Xt+1 to its steady state
Xt
Xt−1
= µD thereafter if 1 < µD < µI.
The steady state is unstable and the disinflation policy is infeasible.
Proposition D.1. The equality X0
X−1 =
X2
X1 holds in the post-disinflation state
only in the special case ofµD= 1.
Proof. The ratio of the wage index in period t and its first lag is given by
Wt Wt−1 =[ 1 2X 1−ε t +12X 1−ε t−1] 1 1−ε [12X1−ε t−1 + 1 2X 1−ε t−2] 1 1−ε (D.1) Setting Wt Wt−1
=µD and manipulating yields
Xt Xt−1 = (µ1−ε D (1 + ( Xt−2 Xt−1 )1−ε )−1)1−1ε (D.2)
This implies that we have
X0 X−1 = (µ1−ε D (1 +µ ε−1 I )−1) 1 1−ε (D.3) X1 X0 = (µ1−ε D (1 + ( X−1 X0 )1−ε )−1)1−1ε (D.4) X2 X1 = (µ1−ε D (1 + ( X0 X1 )1−ε )−1)1−1ε (D.5)
etc. I will now show that for a givenεthe equality X0
X−1
= X2
X1 only holds if
µD= 1 and/or ifµD=µI. Plugging (D.4) into (D.5), we have
X2 X1 = (µ1−ε D (1 + (µ 1−ε D (1 + ( X−1 X0 )1−ε )−1)−1 )−1)1−1ε (D.6) Now using (D.3), X2 X1 = (µ1−ε D (1 + (µ 1−ε D (1 + (µ 1−ε D (1 +µ ε−1 I )−1) −1 )−1)−1 )−1)1−1ε (D.7)
µ1−ε D (1 +µ ε−1 I ) =µ 1−ε D (1 + (µ 1−ε D (1 + (µ 1−ε D (1 +µ ε−1 I )−1) −1 )−1)−1 ) (D.8) and finally after some simplification
µ1−ε I + 1 =µ 1−ε D + (1 +µ ε−1 I −µ ε−1 D ) −1 (D.9) For a givenε, there are only two positive real valued solutions to this equa- tion. These areµD= 1 and µD=µI. This proves the above proposition.
Proposition D.2. If1< µD< µI, then the steady state XXtt
−1 =µD
is locally unstable and it is impossible to carry out the disinflation policy. If0< µD<1,
then the steady state Xt
Xt−1 =µD
is locally stable and the disinflation policy can be carried out.
Proof. From equation (D.2) in Proposition D.1 we have that ( Xt Xt−1 )1−ε =µ1−ε D (1 + ( Xt−1 Xt−2 )ε−1 )−1 (D.10) We define ( Xt Xt−1) 1−ε
=rt. Then we have the first order non-linear difference
equation rt=µ1 −ε D −1 + µ1−ε D rt−1 (D.11) This difference equation has a steady state atr=µ1−ε
D orr=−1. A negative
value of r is economically meaningless. Hence, the only economically meaningful steady state isr =µ1−ε
D , which implies that Xt
Xt−1
=µD. Furthermore, notice
that differentiating rt with respect to its one period lag and evaluating the
derivative at the steady state yields a value of µ11−ε D
, the slope of the phase line at the steady state.
Case 1: 1< µD < µI This case corresponds to a partial disinflation policy.
It is illustrated in figure 11. We begin by noticing that there is a natural initial condition forrt, namelyr0, which is predetermined in this setting. Given
this initial value for rt, the time path of rt is divergent as the steady state
is not locally stable. The reason is that the slope of the phase line given by equation (D.11) is smaller than -1 at the SS. Its horizontal asymptote occurs at a negative value for rt. This means that eventually rt < 0 which implies
that the disinflation policy cannot be carried out. However, notice that our discussion here only covers local as opposed to global stability. As the economy
Figure 11: Case 1: Unstable Pattern
moves farther away from the steady state, and hence nonlinearities become more important, it is possible that the economy behaves in an unexpected way. It is for instance possible that it converges on a so-called 2-cycle, i.e. a path on which fluctuations continue forever at the same amplitude. An analysis of global stability, however, is not within the scope of this paper.
Case 2: 0< µD<1 This case corresponds to disinflation policy to a negative
inflation rate. It is illustrated in figure 12. Notice thatrtis locally stable. The
reason is that the slope of the phase line given by equation (D.11) is greater than -1 at the SS. Its horizontal asymptote occurs at a positive value for rt.
In contrast to Case 1, this implies that the time path ofrt will follow a stable
pattern. The disinflation policy can thus be carried out.28.
Intuitively, the driving factor behind these results is the non(log)linearity of the wage index Wt in Xt and Xt−1.
29 This non(log)linearity is due to the
fact that labor types are not perfectly complementary but at least partially substitutable in the production function. Simply put, if households in one sector set a lower wage than households in the other, the resulting wage index is not given by the simple average of the two wages - as in the linearized version of
28
As discussed above, notice that we only consider local stability.
29
If the production function were of Cobb-Douglas type, the wage index would be loglinear and labor types would be less easily substituted than in the present case.
Figure 12: Case 2: Stable Pattern
the model - but by some value smaller than it. The reason is that firms will substitute labor types from the low wage sector for labor types from the high wage sector. Furthermore, firms will find it optimal to engage more strongly into labor substitution the greater is the wage gap between the two sectors and the greater is the elasticity of technical substitution between labor types, ε. This labor substituting behavior of firms is the reason, why it is not possible to keep wage growth at target throughout future periods after disinflation to positive rate of inflation. In the following, we will contrast wage setting after (a) a complete disinflation, (b) a disinflation to a positive rate of wage inflation and (c) a disinflation towards a negative rate of wage inflation.
µD= 1 is the case of a complete disinflation. We have shown that a complete
disinflation implies that wages set in the post-disinflation state are constant through time for each sector of households. Furthermore, these wages are equal to the respective wage of each sector one period before disinflation. Therefore, this is the one case in which the policy does not introduce dynamics into the behavior of sectoral wages.
Let us now consider the case when 1 < µD < µI. Before starting the
discussion, it is helpful to take a quick look at Figure 1 in the main text which illustrates the post-disinflation path of the new wage in the framework of the linearized version of the model. The question is then, why we do not observe equivalent patterns when investigating disinflations to positive new inflation targets in the framework of the nonlinear model.
Let us assume that we are investigating a disinflation that implies a strong enough reduction in the rate of wage inflation for the new wage in periodt= 0 to
be forced to fall below the prevailing new wage that was set in periodt=−1.30
In the framework of the linearized model, we would be looking at case 1a in which the new wage set in period zero is lower than in the previous period.
We have chosen the case in which the new inflation target is attained ifX0
takes a value somewhere between X−1 and X−2. Now, for a given X−1 and
X−2, let X0A be the new wage that would attain the new inflation target in
period t = 0 in the absence of labor substitution. Notice that X0A > X−2
implies that sectoral wages are less far apart in periodt= 0 than they were in
t = −1. Consequently, if we do allow for labor substitution in period t = 0, firms substitute less labor away from the high wage sector in periodt= 0 than they did in periodt =−1. This implies that the higher wage gains weight in the wage index in periodt = 0 relative to period t = −1 and thereby exerts additional upward pressure on it. This ’labor substitution effect’ implies that
X0 must lie below X0A, the value it would have taken in the absence of labor
substitution. Therefore, in terms of log deviations from the reference steady state, the new wage in period zero must be low relative to its counterpart in the framework of the linearized model.
Moving on to periodt= 1, the smaller isX0, the greater must agents choose
X1 in order to keep wage growth constant. Furthermore, the higher is X1, i.e.
the greater is the wage gap, the more do firms engage into labor substitution away from the higher wage and the higher doesX1have to be chosen in order to
attain the new inflation target. This implies that the ’labor substitution effect’ that put downward pressure on X0 now exerts upward pressure on X1. And
it exerts more upward pressure onX1, the more downward pressure it exerted
onX0. The fluctuations in the new wage are therefore self-enforcing with the
result that the ratio Xt
Xt−1
falls over time throughout even periods and increases over time throughout odd periods until the new wage hits its zero lower bound in even periods.
0 < µD < 1 is the rather unrealistic case of a reduction in wage inflation
towards a negative inflation rate. Here, the economy is saddlepath stable. The ratio Xt
Xt−1 converges towards its post-disinflation steady state
Xt
Xt−1 =µD. No-
tice that this result follows from the above argument as the magnitude of the reduction in the inflation rate is so great that firms engage more strongly into labor substitution in periodt= 0 than they did in periodt=−1.
30